Model of the Newtonian cosmology: Symmetries, invariant and partially invariant solutions

doi:10.1016/j.cnsns.2016.02.029

Communications in Nonlinear Science and Numerical Simulation

Volume 39, October 2016, Pages 248-251

https://doi.org/10.1016/j.cnsns.2016.02.029 Get rights and content

Highlights

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In our paper symmetry group allowed a given model is computed and given group theoretical justification of the “Hubble’s law”.

Abstract

Symmetry group of the equation system of ideal nonrelativistic self-gravitating fluid with zero pressure is calculated. Submodel invariant under the subgroup of rotations SO(3) is built and symmetry group of the factorsystem is calculated. A particular analytical invariant solution of the factorsystem is obtained.

Introduction

Model of the Newtonian cosmology is basic in the study of large-scale structure of the Universe [1], [2], [3], [4], [5], [6], [7]. The model is a system of equations of ideal nonrelativistic self-gravitating fluid with zero pressure, density ρ, velocity $\vec{v}$ and gravitational potential Φ [2] $\begin{matrix} {\begin{matrix} \frac{\partial ρ}{\partial t} + \nabla . (ρ \vec{v}) = 0, \\ \frac{\partial \vec{v}}{\partial t} + (\vec{v} . \nabla) \vec{v} + \nabla (Φ) = 0, \\ Δ Φ = 4 π γ ρ . \end{matrix} \end{matrix}$

The first equation is the equation of continuity, the second – Euler equation, the third – Poisson equation, Δ – Laplace operator, γ is the gravitational constant. The present model is nonrelativistic, but adequately describes the evolution of inhomogeneities in the density of matter in the Universe, because the peculiar velocity and gravitational potential, as is known, do not reach relativistic values (for example, see a detailed discussion in the review [1]. At the present time various approximations of the solution of system (1) are well studied, the best known of them is Zeldovich “pancakes” model [1], [3]. The aim of our work is systematically study the system of equations (1) by methods of group analysis that will yield new exact analytical solutions that not only going beyond perturbation theory but also useful for testing numerical methods.

Due to the symmetry in the model of Newtonian cosmology one can apply the SUBMODELS program, similar to Ovsyannikov’s program in gas dynamics. So, the system (1) will present as a “big model” [10]. The program SUBMODELS allows in principle to obtain all possible invariant and partially invariant solutions of the “big” model and to analyze their physical meaning. This, in our opinion, is the advantage of the methodology of group analysis compared with other methods of mathematical modeling.

Section snippets

Lie point symmetries

Rewrite the system (1) in Cartesian coordinates in dimensionless variables: $\begin{matrix} {\begin{matrix} ρ_{t} + ρ (u_{x} + v_{y} + ω_{z}) + u ρ_{x} + v ρ_{y} + ω ρ_{z} = 0, \\ u_{t} + u u_{x} + v u_{y} + ω u_{z} + Φ_{x} = 0, \\ v_{t} + u v_{x} + v v_{y} + ω v_{z} + Φ_{y} = 0, \\ ω_{t} + u ω_{x} + v ω_{y} + ω ω_{z} + Φ_{z} = 0, \\ Φ_{x x} + Φ_{y y} + Φ_{z z} = ρ, \end{matrix} \end{matrix}$ where x, y, z – Cartesian coordinates, t – time, u, v, ω – the velocity components.

Generator of the group will be sought in the form $\begin{matrix} \hat{X} = ξ^{(x)} \partial_{x} + ξ^{(y)} \partial_{y} + ξ^{(z)} \partial_{z} + ξ^{(t)} \partial_{t} + η^{(ρ)} \partial_{ρ} + η^{(Φ)} \partial_{Φ} + η^{(u)} \partial_{u} + η^{(v)} \partial_{v} + η^{(ω)} \partial_{ω}, \end{matrix}$ where ξ and η are components of the tangent vector field, ∂ – operator of differentiation in the corresponding variable.

Invariant submodel SO(3)

Invariants of the group SO(3) are $\begin{matrix} Φ, ρ, t, r = \sqrt{x^{2} + y^{2} + z^{2}}, | \vec{v} | = \sqrt{u^{2} + v^{2} + ω^{2}} \equiv U, s = \vec{r} \cdot \vec{v} . \end{matrix}$ In view of the fact that $\vec{v} = U \frac{\vec{r}}{r}$ [11], invariant solution should be sought in the form $\begin{matrix} \vec{v} = \frac{\vec{r}}{r} U (t, r), ρ = ρ (t, r), Φ = Φ (t, r) . \end{matrix}$ Substituting (3) in (2) one can obtain a factorsystem $\begin{matrix} {\begin{matrix} U_{t} + U U_{r} + Φ_{r} = 0, \\ ρ_{t} + U ρ_{r} + ρ (\frac{2 U}{r} + U_{r}) = 0, \\ \frac{2 Φ_{r}}{r} + Φ_{r r} = ρ . \end{matrix} \end{matrix}$ Factorsystem (4) allows a 3-dimensional algebra, say, L₃: $\begin{matrix} {\begin{matrix} {\hat{X}}_{1} = 2 Φ \partial_{Φ} + U \partial_{U} + r \partial_{r}, \\ {\hat{X}}_{2} = - 2 ρ \partial_{ρ} + r \partial_{r} + t \partial_{t}, \\ {\hat{X}}_{3} = \partial_{t}, \end{matrix} \end{matrix}$ and infinite-dimensional algebra with generator ${\hat{X}}_{\infty} = F_{1} (t) \partial_{Φ},$ where F₁(t) an arbitrary function of time t.

Conclusion

This paper and [13] initiate the series of papers devoted to the study of invariant and partially invariant solutions of the model of Newtonian cosmology. In subsequent papers we intend to consider the following problems: building all invariant solutions of spherically symmetric submodel SO(3); building partially invariant solution of such “Ovsyannikov’s vortex” [14]; calculation optimal system subalgebras of 13-dimensional Lie algebra, admitted by the “big model” [10].

Acknowledgments

We are grateful A.V. Panov and M.M. Kipnis for useful discussions. This work was partially supported by the Ministry of Education of Russia under Grant no. 2807.

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View full text

Model of the Newtonian cosmology: Symmetries, invariant and partially invariant solutions

Highlights

Abstract

Introduction

Section snippets

Lie point symmetries

Invariant submodel SO(3)

Conclusion

Acknowledgments

Large-scale structure of the universe. The Zeldovich approximation and the adhesion model

Phys Uspekhi

Cosmology

Structure and evolution of the universe

The large-scale structure of the universe

The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium

Rev Mod Phys

Approximation methods for non-linear gravitational clustering

Phys Rep